A scaling limit theorem for controlled branching processes with a size-divisible term

TL;DR

This paper proves a weak convergence theorem for controlled branching processes with a new class of controlled variables, extending previous models by including size-divisible terms and dependent immigration, with implications for continuous-state process limits.

Contribution

It introduces a general framework for convergence of controlled branching processes with size-divisible terms, expanding the scope of previous models and analyzing their limit behavior.

Findings

Established weak convergence conditions for controlled branching processes.

Extended models to include size-divisible and size-dependent immigration terms.

Identified the limit process as a continuous-state process with dependent immigration.

Abstract

We establish general sufficient conditions for a sequence of controlled branching processes to converge weakly on the Skorokhod space. We focus on a class of controlled random variables that extends previous results by considering them as a sum of an immigration size-dependent term and a size-divisible term. Our assumptions are established in terms of the probability generating functions of the offspring and control distributions, distinguishing in this latter case between the immigration and the size-divisible parts. The limit process is a continuous-state process with dependent immigration.